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Dec15-05, 05:15 PM
P: 30
Sorry, but I'm going to have another go at this, since I would like to be able to explain this stuff without making people more confused.

A first-order theory (according to one definition, at least) is a set of wffs closed under logical consequence.

Truth and falsehood are not defined in terms of theories, but in terms of interpretations. We do not say that a wff is true for a theory, only that it belongs to (or is provable) in that theory. We can say that a wff is true for some interpretation though.

If some wff [itex]\phi[/itex] is undecidable in a theory, then there will be an model of that theory (an interpretation which makes all the wffs in the theory true) for which [itex]\phi[/itex] is true and a model for which [itex]\phi[/itex] is false. This follows from the theorem that all consistent theories have models.

In your example, if you have an interpretation consisting of a domain with five elements, and a wff [itex]\phi(x)[/itex] defining a subset of this domain with three elements, then the negation of that wff will define the set's complement. There is no mention of theories here, so it doesn't make sense to talk about the wff being decidable. This is why I didn't see the point of your example.